June 26th
Today I learned a proof of the Hilbert basis theorem. Here is the main attraction.
Theorem. Fix $R$ a Noetherian ring. Then the polynomial ring $R[x]$ is also Noetherian.
At a high level, this proof attempts to emulate the proof that $K[x]$ is a principal ideal domain when $K$ is a field. Namely, we are going to attempt to do Euclidean division to show that $R[x]$ is "small enough'' to be Noetherian.\todo{maybe grobner; maybe R[[x]]}